Chapter Two Budge Budg t e ar t y and Other Constr Cons ain tr ts ain on Choice

Transcription

1 Chapter Two Budgetary and OtherConstraints on Budgetary and Other Constraints on Choice

2 Consumption Choice Sets A consumption choice set is the collection of all consumption choices available to the consumer. What constrains consumption choice? Budgetary, time and other resource limitations.

3 Budget Constraints A consumption bundle containing x 1 units of commodity 1, x 2 units of commodity 2 and so on up to x n units of commodity n is denoted by the vector (x 1, x 2,, x n ). Commodity prices are p 1, p 2,, p n.

4 Budget Constraints Q: When is a consumption bundle (x 1,, x n ) affordable at given prices p 1,, p n? ( 1 n ) g p p 1 p n

5 Budget Constraints Q: When is a bundle (x 1,, x n ) affordable at prices p 1,, p n? A: When p 1 x p n x n m where m is the consumer s (disposable) income.

6 Budget Constraints The bundles that are only just affordable form the consumer s budget constraint. This is the set { (x 1,,x n ) x 1 0,, x n and p 1 x p n x n m }.

7 Budget Constraints The consumer s budget set is the set of all affordable bundles; B(p 1,, p n, m) = { (x 1,, x n ) x 1 0,, x n 0 and p 1 x p n x n m } The budget constraint is the upper boundary of the budget set.

8 x 2 Budget Set and Constraint for Two Commodities Budget constraint is m /p 2 p 1 x 1 +px 2 2 = m. m /p 1 x 1

9 x 2 Budget Set and Constraint for Two Commodities Budget constraint is m /p 2 p 1 x 1 +px 2 2 = m. m /p 1 x 1

10 x 2 Budget Set and Constraint for Two Commodities Budget constraint is m /p 2 p 1 x 1 +px 2 2 = m. Just affordable m /p 1 x 1

11 x 2 Budget Set and Constraint for Two Commodities Budget constraint is m /p 2 p 1 x 1 +px 2 2 = m. Not affordable Just affordable m /p 1 x 1

12 x 2 Budget Set and Constraint for Two Commodities Budget constraint is m /p 2 p 1 x 1 +px 2 2 = m. Not affordable Just affordable Affordable m /p 1 x 1

13 x 2 Budget Set and Constraint for Two Commodities Budget constraint is m /p 2 p 1 x 1 +px 2 2 = m. Budget Set the collection of all affordable bundles. m /p 1 x 1

14 x 2 Budget Set and Constraint for Two Commodities p 1 x 1 + p 2 x 2 = m is m /p 2 x 2 = -(p 1 /p 2 )x 1 + m/p 2 so slope is -p 1 /p 2. Budget Set m /p 1 x 1

15 Budget Constraints If n = 3 what do the budget constraint and the budget set look like?

16 Budget Constraint for Three Commodities x 2 m /p 2 p 1 x 1 + p 2 x 2 + p 3 x 3 = m m /p 3 x x33 m /p 1 x 1

17 Budget Set for Three Commodities x 2 { (x 1,x 2,x 3 ) x 1 0, x 2 0, x 3 0 and m /p 2 p 1 x 1 +p 2 x 2 +p 3 x 3 m} m /p 2 m /p 3 x x33 m /p 1 x 1

18 Budget Constraints For n = 2 and x 1 on the horizontal axis, the constraint s slope is p 1 /p 2. What does it mean? x 2 p p 1 2 x 1 m p 2

19 Budget Constraints For n = 2 and x 1 on the horizontal axis, the constraint s slope is p 1 /p 2. What does it mean? x 2 p p 1 2 x 1 m p 2 Increasing x 1 by 1 must reduce x 2 by p 1 /p 2.

20 Budget Constraints x 2 Slope is -p 1 /p 2 -p 1 /p 2 +1 x 1

21 x 2 Budget Constraints 2 Opp. cost of an extra unit of commodity 1 is p 1 /p 2 units foregone of commodity 2. -p 1 /p 2 +1 x 1

22 x 2 Budget Constraints 2 Opp. cost of an extra unit of commodity 1 is p 1 /p 2 units foregone of commodity 2. And the opp. cost of an extra +1 unit of commodity 2 is -p 2 /p 1 p 2 /p 1 units foregone of commodity 1. x 1

23 Budget Sets & Constraints; Income and Price Changes The budget constraint and budget set depend upon prices and income. What happens as prices or income change?

24 x 2 How do the budget set and budget constraint tchange as income m increases? Original budget set x 1

25 Higher income gives more choice x 2 New affordable consumption choices Original and new budget constraints are parallel (same Original slope). budget set x 1

26 x 2 How do the budget set and budget constraint tchange as income m decreases? Original budget set x 1

27 x 2 How do the budget set and budget constraint tchange as income m New, smaller budget set decreases? Consumption bundles that t are no longer affordable. Old and new constraints are parallel. x 1

28 Budget Constraints Income Changes Increases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice.

29 Budget Constraints Income Changes Increases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice. Decreases in income m shift the constraint inward in a parallel manner, thereby shrinking the budget set and reducing choice.

30 Budget Constraints Income Changes No original choice is lost and new choices are added when income increases, so higher income cannot make a consumer worse off. An income decrease may (typically will) make the consumer worse off.

31 Budget Constraints Price Changes What happens if just one price decreases? Suppose p 1 decreases.

32 How do the budget set and budget constraint tchange as p 1 decreases x 2 from p 1 to p 1?? m/p 2 -p 1 /p 2 Original budget set m/p 1 m/p 1 x 1

33 How do the budget set and budget constraint tchange as p 1 decreases x 2 from p 1 to p 1?? m/p 2 New affordable choices -p 1 /p 2 Original budget set m/p 1 m/p 1 x 1

34 How do the budget set and budget constraint tchange as p 1 decreases x 2 from p 1 to p 1?? m/p 2 New affordable choices -p 1 /p 2 Original budget set Budget constraint pivots; slope flattens from -p 1 /p 2 to -p -p 1 /p 2 1 /p 2 m/p 1 m/p 1 x 1

35 Budget Constraints Price Changes Reducing the price of one commodity pivots the constraint outward. No old choice is lost and new choices are added, so reducing one price cannot make the consumer worse off.

36 Budget Constraints Price Changes Similarly, increasing one price pivots the constraint inwards, reduces choice and may (typically will) make the consumer worse off.

37 Uniform Ad Valorem Sales Taxes An ad valorem sales tax levied at a rate of 5% increases all prices by 5%, from p to (1+0 05)p = 1 05p. An ad valorem sales tax levied at a rate of t increases all prices by tp from p to (1+t)p. A uniform sales tax is applied uniformly to all commodities.

38 Uniform Ad Valorem Sales Taxes A uniform sales tax levied at rate t changes the constraint from p 1 x 1 + p 2 x 2 = m to (1+t)p 1 x 1 + (1+t)p 2 x 2 = m

39 Uniform Ad Valorem Sales Taxes A uniform sales tax levied at rate t changes the constraint from p 1 x 1 + p 2 x 2 = m to i.e. (1+t)p 1 x 1 + (1+t)p 2 x 2 = m p 1 x 1 + p 2 x 2 = m/(1+t).

40 Uniform Ad Valorem Sales Taxes x 2 m p p 2 p 1 x 1 + p 2 x 2 = m x 1 m 1 p 1

41 Uniform Ad Valorem Sales Taxes x 2 m p 1 x 1 + p 2 x 2 = m p 2 m p 1 x 1 + p 2 x 2 = m/(1+t) ( tp ) 1 2 m ( tp ) 1 1 m p 1 x 1

42 Uniform Ad Valorem Sales Taxes x 2 m p 2 m ( tp ) 1 2 Equivalent income loss is m m t 1 t 1 t m m m x 1 ( tp ) p 1 1 1

43 Uniform Ad Valorem Sales Taxes x 2 A uniform ad valorem sales tax levied at rate t m is equivalent to an income p 2 tax levied at rate t m 1 tt. ( tp ) 1 2 m m x 1 ( tp ) p 1 1 1

44 The Food Stamp Program Food stamps are coupons that can be legally exchanged only for food. How does a commodity specific gift such as a food stamp alter a family s budget constraint?

45 The Food Stamp Program Suppose m = $100, p F = $1 and the price of other goods is p G = $1. p G The budget constraint is then F + G =100.

46 G The Food Stamp Program F + G = 100; before stamps F

47 G The Food Stamp Program F + G = 100: before stamps F

48 G The Food Stamp Program F + G = 100: before stamps. 100 Budget set after 40 food stamps issued F

49 G The Food Stamp Program F + G = 100: before stamps. 100 Budget set after 40 food stamps issued. The family s budget set is enlarged F

50 The Food Stamp Program What if food stamps can be traded on a black market for $0.50 each?

51 The Food Stamp Program G F + G = 100: before stamps Budget constraint after 40 food stamps issued. Budget constraint with black market trading F

52 The Food Stamp Program G F + G = 100: before stamps Budget constraint after 40 food stamps issued. Black market trading makes the budget set larger again F

53 Budget Constraints Relative Prices Numeraire means unit of account. Suppose prices and income are measured in dollars. Say p 1 =$2, p 2 =$3, m = $12. Then the constraint is 2x 1 + 3x 2 = 12.

54 Budget Constraints Relative Prices If prices and income are measured in cents, then p 1 =200, p 2 =300, m=1200 and the constraint is 200x x 2 = 1200, the same as 2x 1 + 3x 2 = 12. Changing the numeraire changes neither the budget constraint nor the budget set.

55 Budget Constraints Relative Prices The constraint tfor p 1 =2, p 2 =3, m=12 2x 1 + 3x 2 = 12 is also 1.x 1 + (3/2)x 2 = 6, the constraint for p 1 =1, p 2 =3/2, m=6. Setting p 1 =1 makes commodity 1 the numeraire and defines all prices relative to p 1 ; e.g. 3/2 is the price of commodity 2 relative lti to the price of commodity 1.

56 Budget Constraints Relative Prices Any commodity can be chosen as the numeraire without changing gthe budget set or the budget constraint.

57 Budget Constraints Relative Prices p 1 =2, p 2 =3 and p 3 =6 price of commodity 2 relative to commodity 1 is 3/2, price of commodity 3 relative to commodity 1 is 3. Relative prices are the rates of exchange of commodities 2 and 3 for units of commodity 1.

58 Shapes of Budget Constraints Q: What makes a budget constraint a straight line? A: A straight line has a constant slope and the constraint is p 1 x p n x n = m so if prices are constants then a constraint is a straight line.

59 Shapes of Budget Constraints But what if prices are not constants? E.g. bulk buying discounts, or price penalties for buying too much. Then constraints will be curved.

60 Shapes of Budget Constraints Quantity Discounts Suppose p 2 is constant at $1 but that p 1 =$2 for 0 x 1 20 and p 1 =$1 for x 1 >20. 1 p 1 1

61 Shapes of Budget Constraints Quantity Discounts Suppose p 2 is constant at $1 but that p 1 =$2 for 0 x 1 20 and p 1 =$1 for x 1 >20. Then the constraint s slope is 2, for 0 x 1 20 { p 1 /p 2 = 1, for x 1 > 20 and the constraint is

62 Shapes of Budget Constraints with a Quantity Discount x 2 m = $ Slope = - 2 / 1 = - 2 (p 1 =2, p 2 =1) Slope = - 1/ 1 = - 1 (p 1 =1, p 2 =1) x 1

63 Shapes of Budget Constraints with a Quantity Discount x 2 m = $ Slope = - 2 / 1 = - 2 (p 1 =2, p 2 =1) Slope = - 1/ 1 = - 1 (p 1 =1, p 2 =1) x 1

64 Shapes of Budget Constraints with a Quantity Discount m = $100 x 2 m = $ Budget Constraint Budget Set x 1

65 Shapes of Budget Constraints with x 2 a Quantity Penalty Budget Constraint Budget Set x 1

66 Shapes of Budget Constraints One Price Negative Commodity 1 is stinky garbage. You are paid $2 per unit to accept it; i.e. p 1 = $2. p 2 = $1. Income, other than from accepting commodity 1, is m = $10. Then the constraint is 2x 1 + x 2 = 10 or x 2 = 2x

67 Shapes of Budget Constraints One Price Negative x 2 x 2 = 2x Budget constraint s slope is -p 1 /p 2 = -(-2)/1 = x 1

68 Shapes of Budget Constraints One Price Negative x 2 10 Budget set is all bundles for which x 1 0, x 2 0 and x 2 2x x 1

69 More General Choice Sets Choices are usually constrained by more than a budget; e.g. time constraints and other resources constraints. A bundle is available only if it meets every A bundle is available only if it meets every constraint.

70 More General Choice Sets Other Stuff At least 10 units of food must be eaten to survive 10 Food

71 More General Choice Sets Other Stuff Choice is also budget constrained. Budget Set 10 Food

72 More General Choice Sets Other Stuff Choice is further restricted by a time constraint. 10 Food

73 More General Choice Sets So what is the choice set?

74 More General Choice Sets Other Stuff 10 Food

75 More General Choice Sets Other Stuff 10 Food

76 More General Choice Sets Other Stuff The choice set is the intersection of all of the constraint t sets. 10 Food